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Derivative of Polynomials: Product Rule
Related Topics
Wize University Calculus 1 Textbook > Derivatives
Differentiation Laws
1 Activity
Wize University Calculus 1 Textbook > Derivatives
The Power Rule (Derivative of Polynomials)
3 Activities
Find the derivative of
f
(
x
)
=
(
x
3
/
2
+
2
x
)
(
x
7
/
4
+
x
)
f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
f
(
x
)
=
(
x
3/2
+
x
2
)
(
x
7/4
+
x
)
f
′
(
x
)
=
21
8
x
5
4
+
3
2
x
1
2
−
7
2
x
5
4
−
2
x
2
f'\left(x\right)=\frac{21}{8}x^{\frac{5}{4}}+\frac{3}{2}x^{\frac{1}{2}}-\frac{7}{2}x^{\frac{5}{4}}-\frac{2}{x^2}
f
′
(
x
)
=
8
21
x
4
5
+
2
3
x
2
1
−
2
7
x
4
5
−
x
2
2
f
′
(
x
)
=
−
1
4
x
9
4
+
1
2
x
3
2
+
3
2
x
−
3
8
−
4
x
f'\left(x\right)=-\frac{1}{4}x^{\frac{9}{4}}+\frac{1}{2}x^{\frac{3}{2}}+\frac{3}{2}x^{-\frac{3}{8}}-\frac{4}{x}
f
′
(
x
)
=
−
4
1
x
4
9
+
2
1
x
2
3
+
2
3
x
−
8
3
−
x
4
f
′
(
x
)
=
13
4
x
9
/
4
+
5
2
x
3
/
2
+
3
2
x
1
/
4
f'(x)=\frac{13}{4}x^{9/4}+\frac{5}{2}x^{3/2}+\frac{3}{2x^{1/4}}
f
′
(
x
)
=
4
13
x
9/4
+
2
5
x
3/2
+
2
x
1/4
3
I don't know
Check Submission
More Differentiation Laws Questions:
Derivative of Polynomials: Product Rule
Find the derivative of
f
(
x
)
=
(
x
3
/
2
+
2
x
)
(
x
7
/
4
+
x
)
f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
f
(
x
)
=
(
x
3/2
+
x
2
)
(
x
7/4
+
x
)
Differentiation laws
What is the derivative of the function
f
(
x
)
=
x
3
+
1
x
2
f(x)=\frac{x^3+1}{x^2}
f
(
x
)
=
x
2
x
3
+
1
?
Differentiation: Piecewise Differentiable Function
Find
A
A
A
and
B
B
B
for which
f
(
x
)
f\left(x\right)
f
(
x
)
is differentiable everywhere. Answers are in the form
(
A
,
B
)
\left(A,B\right)
(
A
,
B
)
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
Differentiation: Piecewise Differentiable Function
Find
A
A
A
and
B
B
B
for which
f
(
x
)
f\left(x\right)
f
(
x
)
is differentiable everywhere.
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
Derivative of Polynomials: Product Rule
Find the derivative of
f
(
x
)
=
(
x
3
/
2
+
2
x
)
(
x
7
/
4
+
x
)
f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
f
(
x
)
=
(
x
3/2
+
x
2
)
(
x
7/4
+
x
)
Differentiation: Piecewise Differentiable Function
Find
A
A
A
and
B
B
B
for which
f
(
x
)
f\left(x\right)
f
(
x
)
is differentiable everywhere.
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
Differentiation: Piecewise Differentiable Function
Find
A
A
A
and
B
B
B
for which
f
(
x
)
f\left(x\right)
f
(
x
)
is differentiable everywhere.
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
Differentiation: Piecewise Differentiable Function
Find
A
A
A
and
B
B
B
for which
f
(
x
)
f\left(x\right)
f
(
x
)
is differentiable everywhere.
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
Derivatives: Logarithmic Functions
Compute the derivative of
f
(
x
)
=
x
x
+
1
f(x) = x^{x + 1}
f
(
x
)
=
x
x
+
1
. Remember that
log
x
=
log
e
x
=
ln
x
\log x = \log_e x = \ln x
lo
g
x
=
lo
g
e
x
=
ln
x
.
The chain rule
Find the derivative of
h
(
x
)
=
log
(
cos
(
x
)
)
h(x) = \log(\cos(x))
h
(
x
)
=
lo
g
(
cos
(
x
))
. Remember that
log
x
=
log
e
x
=
ln
x
\log x = \log_e x = \ln x
lo
g
x
=
lo
g
e
x
=
ln
x
.
The Quotient Rule
Find the derivative of
g
(
x
)
=
x
2
−
5
2
x
+
1
g(x) = \frac{x^2 - 5}{2x + 1}
g
(
x
)
=
2
x
+
1
x
2
−
5
The product rule
Find the derivative of
f
(
x
)
=
x
3
e
x
f(x) = x^3 e^x
f
(
x
)
=
x
3
e
x
Differentiation: Piecewise Differentiable Function
Find
A
A
A
and
B
B
B
for which
f
(
x
)
f\left(x\right)
f
(
x
)
is differentiable everywhere.
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
f
(
x
)
=
{
x
e
x
2
+
1
,
if
x
≥
1
A
x
+
B
,
if
x
<
1
Differentiation laws
What is the derivative of the function
f
(
x
)
=
x
3
+
1
x
2
f(x)=\frac{x^3+1}{x^2}
f
(
x
)
=
x
2
x
3
+
1
?
Is
f
(
x
)
f(x)
f
(
x
)
differentiable at
x
=
1
x=1
x
=
1
? If so, find
f
′
(
1
)
f'(1)
f
′
(
1
)
.
f
(
x
)
=
{
x
+
1
if
x
<
1
1
2
x
2
+
3
2
if
x
≥
1
f(x) = \begin{cases} x+1 & \text{if } x < 1 \\ \frac{1}{2}x^2 + \frac{3}{2} & \text{if } x \geq 1 \end{cases}
f
(
x
)
=
{
x
+
1
2
1
x
2
+
2
3
if
x
<
1
if
x
≥
1
Differential Laws: nth Derivatives
If
y
=
(
10
e
+
1
)
10
y=\left(10e+1\right)^{10}
y
=
(
10
e
+
1
)
10
, find the 9
th
derivative of
y
y
y
.
If
f
(
x
)
f(x)
f
(
x
)
is differentiable everywhere, find
A
A
A
and
B
B
B
.
f
(
x
)
=
{
x
2
+
1
if
x
≥
0
A
x
+
B
if
x
<
0
f(x)=\left\{ \begin{array}{ll} \displaystyle x^2+1\quad\quad\,\,\,\,\,\text{if}\,x\geq 0 \\ Ax+B\quad\,\,\,\,\,\,\,\text{if}\,\, x<0\\ \end{array} \right.
f
(
x
)
=
{
x
2
+
1
if
x
≥
0
A
x
+
B
if
x
<
0
More The Power Rule (Derivative of Polynomials) Questions:
Higher Order Derivatives: The Power Rule
If
f
(
x
)
=
x
10
f\left(x\right)=x^{10}
f
(
x
)
=
x
10
find
f
(
9
)
(
x
)
f^{\left(9\right)}\left(x\right)
f
(
9
)
(
x
)
.
(i.e. the 9th derivative)
Derivative of Polynomials: Product Rule
Find the derivative of
f
(
x
)
=
(
x
3
/
2
+
2
x
)
(
x
7
/
4
+
x
)
f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
f
(
x
)
=
(
x
3/2
+
x
2
)
(
x
7/4
+
x
)
The Power Rule: Derivatives of Polynomials
Find the derivative of
f
(
x
)
=
x
3
x
f\left(x\right)=\dfrac{x^3}{\sqrt{x}}
f
(
x
)
=
x
x
3
Derivative of Polynomials: Product Rule
Find the derivative of
f
(
x
)
=
(
x
3
/
2
+
2
x
)
(
x
7
/
4
+
x
)
f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
f
(
x
)
=
(
x
3/2
+
x
2
)
(
x
7/4
+
x
)
Higher Order Derivatives: The Power Rule
If
f
(
x
)
=
x
10
f\left(x\right)=x^{10}
f
(
x
)
=
x
10
find
f
(
9
)
(
x
)
f^{\left(9\right)}\left(x\right)
f
(
9
)
(
x
)
.
(i.e. the 9th derivative)
Find the derivative of
f
(
x
)
=
x
3
x
f\left(x\right)=\frac{x^3}{\sqrt{x}}
f
(
x
)
=
x
x
3
Derivative of Polynomials
Suppose
f
is a function which satisfies
f
′
(
0
)
=
π
and
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
f'(0)=\pi \text{ and }f(x+y)=f(x)+f(y)
f
′
(
0
)
=
π
and
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
for all
x
and
y
. True or false:
f
(
x
)
=
π
x
f(x)=\pi x
f
(
x
)
=
π
x
satisfies this function.
The Power Rule: Derivatives of Polynomials
Find the derivative of
f
(
x
)
=
x
3
x
f\left(x\right)=\dfrac{x^3}{\sqrt{x}}
f
(
x
)
=
x
x
3
If 𝑓(𝑥) = 𝑥
10
, find 𝑓
(9)
(𝑥).
(i.e. the 9th derivative)
Find the derivative of
f
(
x
)
=
x
3
x
f\left(x\right)=\frac{x^3}{\sqrt{x}}
f
(
x
)
=
x
x
3